Heat Transfer and Melting in Subglacial Basaltic Volcanic Eruptions: Implications for Volcanic Deposit Morphology and Meltwater Volumes

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Heat transfer and melting in subglacial basaltic volcanic eruptions: implications for volcanic deposit morphology and meltwater volumes

LIONEL WILSON 1'2 & JAMES W. HEAD, III 2

1 Department of Environmental Science, Lancaster University, Lancaster LA1 4YQ, UK (e-mail: L. [email protected]) 2 Department of Geological Sciences, Brown University, Providence, RI 02912, USA

Abstract: Subglacial volcanic eruptions can generate large volumes of meltwater that is stored and transported beneath glaciers and released catastrophically in j6kulhlaups. At typical basaltic dyke propagation speeds, the high strain rate at a dyke tip causes ice to behave as a brittle solid; dykes can overshoot a rock-ice interface to intrude through 20-30% of the thickness of the overlying ice. The very large surface area of the dyke sides causes rapid melting of ice and subsequent collapse of the dyke to form a basal rubble pile. Magma can also be intruded at the substrate-ice interface as a sill, spreading sideways more efficiently than a subaerial flow, and also producing efficient and widespread heat transfer. Both intrusion mechanisms may lead to the early abundance of meltwater sometimes observed in Icelandic subglacial eruptions. If meltwater is retained above a sill, continuous melting of adjacent and overlying ice by hot convecting meltwater occurs. At typical sill pressures under more than 300 m ice thickness, magmatic CO2 gas bubbles form c. 25 vol% of the pressurized magma. If water drains and contact with the atmosphere is established, the pressure decreases dramatically unless the overlying ice subsides rapidly into the vacated space. If it does not, further CO2 exsolution plus the onset of H20 exsolution has the potential to cause explosive fragmentation, i.e. a fire-fountain that forms at the dyke-sill connection, enhancing melting and creating another candidate pulse of meltwater. The now effectively subaerial magma body becomes thicker, narrower, and flows faster so that marginal meltwater drainage channels become available. If the ice overburden thickness is much less than c. 300m the entire sill injection process may involve explosive magma fragmentation. Thus, there should be major differences between subglacial eruptions under local or alpine glaciers compared with those under continental-scale glaciers.

Subglacial volcanic eruptions have been studied processes elsewhere on Earth (Mathews 1947; extensively in Iceland (Bj6rnsson 1975; Allen Skilling 1994; Smellie & Skilling 1994; Chapman 1980; Gudmundsson & Bj6rnsson 1991; Gud- et al. 2000) and on Mars (Allen 1979; Hodges & mundsson et al. 1997; Johannesson & Saemunds- Moore 1994; Head & Wilson 2002). son 1998) due to the ongoing nature of the Dykes represent the propagation, both later- process and the many beautifully exposed land- ally and vertically, of sub-vertical magma-filled forms and deposits. Of particular interest is the cracks from crustal or subcrustal reservoirs into generation of large volumes of meltwater, its the surrounding area. Dykes may propagate storage and transport below the glaciers, and the to the surface to cause eruptions; may propa- catastrophic meltwater release at glacial margins gate to the near-surface to set up stress fields, to produce j6kulhlaups (Bj6rnsson 1975, 1992). which under suitable conditions result in graben Documentation of the products and landforms (Mastin & Pollard 1988; Rubin 1992); or may resulting from these eruptions (Bj6rnsson 1975; stall and cool in the crust at depths too great to Allen 1980; Gr6nvold & Johannesson 1983; produce visible indications of their presence. The Gudmundsson et al. 1997; Johannesson & Sae- latter includes the possibility that they may cease mundsson 1998) and continuing study of active vertical propagation at some relatively shallow examples (Gudmundsson et al. 1997), together depth and then spread sideways to produce sills. with the development of qualitative and quanti- This process is encouraged if the least principal tative models of the processes (Einarsson 1966; stress ceases to be horizontal and becomes Gudmundsson et al. 1997; H6skuldsson & vertical. The discontinuity in density and other Sparks 1997; Hickson 2000; Smellie 2000), has material properties provided by the contact led to the recognition of candidates for these between a glacier or ice-cap and the underlying

From: SMELLIE, J. L. & CHAPMAN, M. G. (eds) 2002. Volcano-Ice Interaction on Earth and Mars. Geological Society, London, Special Publications, 202, 5-26. 0305-8719/02/$15.00 © The Geological Society of London 2002. Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

6 L. WILSON & J. W. HEAD rocks may also be a trigger for such activity, and magma reservoirs are typically c. 3 MPa (Parfitt subglacial eruptions are likely to begin with the 1991) and for reservoir depths of a few kilo- intrusion of a sill at the rock-ice boundary. metres these correspond to similar pressure Commonly, a subaerial basaltic eruption is gradients of c. 1000 Pa m -1. The consequence is initially manifested as a curtain of fire along a that the magma in mafic dykes with typical fissure tens to hundreds of metres long which widths of c. 1 m propagates upward at speeds of marks the surface trace of the dyke. Cooling c. 1 m s -1 (Wilson & Head 1981). The strain rates along the narrow parts of the dyke (Wilson & near the dyke tips implied by these speeds are Head 1988) causes localization of extrusion c. 1 s -I, about seven orders of magnitude larger within a few hours to a few days, and transition than the strain rates at which the surrounding ice to a centralized vent eruption (Head & Wilson can flow plastically given the rheological models 1987; Bruce & Huppert 1989). In submarine (a pseudo-plastic power-law fluid with a yield (Head et al. 1996) and subglacial basaltic erup- strength) proposed by Glen (1952), Nye (1953) tions, a classical initial curtain of fire does not and Paterson (1994). Thus a dyke can easily generally occur because of the inhibition of gas overshoot an ice-rock interface because the ice exsolution due to the pressure of the overlying appears to the propagating crack as a brittle, water or ice. In submarine eruptions, the sup- low-density rock with elastic properties similar pression of gas release continues throughout the to those of the basalt substrate. We show that eruption, but in subglacial eruptions the situa- the amount of ice melting which takes place on tion may become much more complex. Melting the timescale of dyke emplacement may be small of the ice overlying the initial sill may form a enough for the emplacement process to be cavity. As long as the overlying ice does not stable, though subsequent, more extensive ice deform too quickly, the pressure in the cavity melting may lead to collapse of the dyke. may be less than the lithostatic load which acted The pressure distribution in a dyke propagat- on the sill during the early stages of the intrusion ing through an elastic medium is dictated by process, and this may lead to an increase in several requirements that must be met simulta- gas exsolution and magma vesiculation, possibly neously. Most fundamental is that the distribu- resulting in magma fragmentation and some tion of stress across the dyke wall (dictated by form of explosive activity. The overlying ice both the internal pressure distribution and the cover may be completely removed, exposing external stress distribution) must be such as to magma to the pressure of the atmosphere and hold open the sides of the fracture into which leading to more vigorous explosive activity. magma is moving. There must also be a vertical With suitable additions, existing physical pressure gradient in the magma to support the models for the ascent and eruption of magma static weight of the magma, and an additional (Wilson & Head 1981, 1983) can be applied to pressure gradient in the direction of magma subglacial environments. Here we develop some travel to drive the motion against wall friction. simple physical principles for the intrusion of To maximize the flow speed, and hence the mass magma into a glacial cover and assess the impli- and volume fluxes through a dyke of a given cations for eruption behaviour and the nature of shape, the pressure in the propagating tip of the the resulting volcanic deposits and meltwater dyke, Ppt, must decrease to a low value. The release processes. We discuss the conditions theoretical ideal tip pressure is zero, but Rubin under which hyaloclastites and lava breccias (1993) suggested that tip pressure would in fact form, and show how either lava flow units or sill- be no smaller than the pressure at which the like bodies can form at the base of the ice, most soluble volatile species which the magma depending on the melting rate and behaviour of contains, commonly water, becomes saturated. the ice dictated by its thickness. The argument is that if the pressure falls slightly below the value at which the magma is saturated Subglacial and englacial dyke emplacement in this volatile, more of the volatile exsolves. The solubility function for water in basalt (Wilson & Mafic dykes sourced in crustal magma reservoirs Head 1981) is: are driven upward by magma buoyancy, by the nw = KwP 07 (1) presence of an excess pressure in the reservoir, or by a combination of the two. We shall show in where the constant Kw is 6.8 × 10 -8 if nw is later sections that typical mafic magmas have expressed as a mass fraction and P is the bulk densities smaller than those of their host pressure in Pascals. If the magma contains 0.25 rocks by Ap =c. 200kgm -3, so that the buoy- mass% water, a plausible value for a mafic ancy pressure gradient acting on them (g Ap =) is magma (Gerlach 1986), n=0.0025 and the c. 2000 Pa m -1 . Excess pressures in crustal mafic saturation pressure, and hence the propagating Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 7 dyke tip pressure, is close to 3.3 MPa; we use this et al. 1977), and that y will lie in the range 500 to value in many subsequent calculations. We note, at most 2000m based on ice cap thicknesses however, that reducing the assumed water under current (up to c. 900m Sigmundsson & content by a factor of two would imply a pres- Einarsson 1992; Einarsson 1994) and glacial sure of 1.2 MPa whereas increasing it by a factor (1000-1500m Einarsson & Albertsson 1988; of two would imply 9.0 MPa. We comment on Geirsdottir & Ericksson 1994; Bourgeois et al. the implications of this later. 1998) conditions. The lithostatic pressure Pr at When a vertically-propagating dyke comes to the top of the relaxed magma reservoir will then rest with its tip at some point below the surface, lie within the extremes of 27 and 86MPa. The the pressure gradient due to the motion will by solubility nc of CO2 in basaltic magmas is given definition have vanished. In general, any excess by (Harris 1981) pressure originally present in the magma reservoir and driving the intrusion will also have nc = Jc + Kc P (2) vanished, though residual pressure gradients where Jc equals 3.4x 10 -6 and Kc equals may still be present if the magma in any part 6 x 10 -12 Pa -1 when nc is expressed as a mass of the dyke system has a non-Newtonian fraction. Assuming a plausible basaltic magma rheology involving a finite yield strength (Parfitt content n t of this volatile, say 0.2 mass% i.e. & Wilson 1994). Figure 1 shows the configura- 0.002 mass fraction, the mass fraction exsolved tion of such a dyke propagating from a reservoir at 27MPa is 0.00183 and at 86MPa is 0.00148. at depth z below the rock-ice interface. A layer At the dyke tip, where the pressure is likely to be of ice of thickness y exists at this location no less than the value during propagation and the tip of the dyke comes to rest at a (Ppt = C. 3.3 MPa), the amount of CO2 exsolved depth x below the ice surface. The density pi of will be ne = (nt - nc) = c. 0.00198. The bulk den- the ice is c. 917kgm -3. The density Pr of the sity/3 of the magma is given by crustal rocks is controlled by their likely origin as a mixture of vesicular lavas and possibly 1//3 = [ne/Pc] + [(1 - ne)/Pm] (3) poorly packed pyroclastics which have under- where Pc is the density of the CO2 given to an gone various kinds of weathering and alteration: adequate approximation by the ideal gas law: we assume a value of 2300 kg m -3, close to that implied by the inversion of seismic data (Hill pc = (mcP)/(QTm). (4) 1969; Zucca et al. 1982; Gudmundsson 1987; Head & Wilson 1992). To estimate the average Here mc is the molecular mass of CO2, magma density between the reservoir and the 43.99 kg kmo1-1, Q is the universal gas constant, trapped tip we recall that the tip pressure is 8.314kJkmol -l K -l, Tm is the magma tempera- likely to be buffered by H20 exsolution so that ture, 1473 K (1200°C), and Pm is the density of the the only exsolved volatile phase will be CO> basaltic magmatic liquid, say 2700 kg m -3. Using present as bubbles of gas or supercritical fluid in these values, the magma bulk density varies from the magma. We assume that z is likely to be 1863 kg m -3 at 3.3 MPa to 2574 kg m -3 at 27 MPa in the range 1 to 3 km, based on the depths of to 2669 kg m -3 at 86 MPa. The mean bulk den- shallow magma reservoirs in Iceland (Bj6rnsson sity, tim, of the magma in the dyke between the pressure in the tip, Ppt, and at the reservoir roof, Pr, is evaluated from

=Pr 9i /3m = (Pr -- Ppt)-1 /3 dP (5a) f~=Ppt Using equations (2) and (3), and defining the 9r convenient constants a = [pmQTm(nt - arc)], b = [mc(1 - nt + Jc) - pmQTmKc],

c = [mcKcl, e = [pm/(2Kc)], Fig. 1. Geometry of an englacial dyke extending to h -~ [(b 2 - 4ac)l/2], within a distance x of the upper surface of an ice layer and of thickness y from a reservoir a distance z below the ice-rock interface. f-= [(pmb)/(2Kch)], Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

8 L. WILSON & J. W. HEAD we find: because the dyke tip pressure during dyke emplacement will be buffered at a different value. /3m = (Pr - Ppt) -1 Equation (1) shows that varying the water content between 0.125% and 1% causes the propa- x [eln{(a + bPr + cPZ)/(a + bPpt + cp2t)} gating tip pressure Ppt to vary between 1.2 MPa and 24MPa. Equation (2) shows that the - fln{[(2cPr + b - h) exsolved amount of CO2 would change by

X (2cPpt q- b + h)]/[(2cP, + b + h) c. 7% as a result. The change in tip pressure is thus somewhat more important than the resulting x (2cPpt + b - h)]}]. (5b) change in CO2 content of the magma. The mean magma densities correponding to Ppt = 1.2 and If the roof of the magma reservoir is at the 24 MPa are 2361 and 2564 kg m -3, respectively, if Pr = 27 MPa level, the mean density of magma in Pr=27MPa, and are 2553 and 2635kgm -3, the dyke will be c. 2390kgm -3 and if the reser- respectively, if Pr=86MPa, typically a 4% voir roof pressure is 86 MPa the mean dyke variation for the smaller Pr and a 1.6% variation magma density will be c. 2567 kg m -3. Whatever for the larger P~. the geometry, therefore, the mean density of the As the tip of the dyke comes to rest, the magma will lie within c. 4% of the value pressure in the gas in the tip cavity will increase /3 = 2480 kgm -3. This value will change if the from the low, buffered value maintained during assumed magma water content is changed, magma motion and will reach a final value Pt.

40 / / / / / l / i / i i / I i i i I /: / 35 -- -- - y =500rn ..... y = 1000 m / ..... y = 1500 rn / ...... ~ = 2000 m ,I 30 ace-rod~ boundary / I . buffered lip pressure / • / • 25 / I o P/MPa i / s o 20 / ,," o S •

dan' • • dF • • • t,~ m

15 ,m,,m' m ROCK 10 ,i i / I ~ n l Ii l ld ...... i

0 0 5 O0 1000 15 O0 2:000

x/m Fig. 2. The pressure, Pt, in the gas pocket in the tip of a dyke after it has been intruded into an ice layer (see Fig. 1) as a function of the depth, x of the tip below the ice surface and the thickness, y of the ice layer. The horizontal broken line indicates the smallest pressure, for the chosen magma volatile content (see text), likely to exist in the dyke tip while it is propagating. The inclined solid line shows the location of the ice-rock interface. Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 9

This pressure can be found by assuming that the before it came to rest: any decrease in pressure hot rocks near the roof of the magma reservoir below this value would lead to additional ex- cannot support significant stresses (whereas the solution of water from the magma. This would cold rocks or ice around the tip of the dyke lead to a decrease in magma density near the can support stresses as large as their mechan- dyke tip but would not greatly change the mean ical strengths). The balance between lithostatic magma density used in the calculation. The (g[ZPr + YPi]) and magma (Pt + g/3[z + {y - x}]) oblique line on the graph shows the boundary stresses at the roof of the magma reservoir (see between dyke tips located within the ice layer Fig. 1) implies, after collecting terms, that (below the oblique line) and those located within the silicate rock crust (above the line). Clearly Pt = g[/3x - (/3 - Pi)Y + (Pr -/3)z]. (6) there is a wide range of conditions under which a Using the densities adopted above, we find dyke could penetrate, and stall within, the ice. (/3 - pi)-- 1333 kgm -3 and (Pr -/3) = 50 kgm -3. The pressure in the dyke tip in excess of the The relatively small value of (/3 - Pr) means that local lithostatic load of the overlying ice, Pe, is Pt is only weakly dependent on the magma equal to (Pt - gpix) and so using equation (6) reservoir depth, z and is controlled mainly by the Pt = g[(/3 - pi)(x - y) + (pr - t3)z]. (7) ice thickness, y and the depth of the dyke tip below the surface, x. Figure 2 shows how Pt Figure 3 shows how Pe varies with x for the varies with x for y = 500, 1000, 1500 and 2000 m. same set of values of y as Figure 2. Physically, Pe The horizontal line on this graph shows the water may be either positive or negative. The bound- pressure of 3.3 MPa, the pressure in the dyke tip ary between dyke tips in ice and dyke tips in

25 • " " " I " " " " I " " • " I • " " "

-- -- - y = 500 m 20 ..... y= lOOO m ..... y = 15oo m ...... y=zooom 15 Ice-rock boundary ,Co

J 10 h

P /MPa 0 ROCK .,, .- " 5 J • J - ! j ,l ~ ~,0'

• • #.i d

i 5 • ### 'r'r

-I0 L c ".' i i i

15 • • i • • • i • • • 0 500 1000 1500 2000 x/m

Fig. 3. The difference in pressure, Pc, between the gas in an intruded dyke tip and the external lithostatic load for the intrusion geometries corresponding to Figure 2. See text for discussion. Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

10 L. WILSON & J. W. HEAD

rock, which corresponds to setting x equals y in buffered value of c. 3.3MPa takes the values equation (7), is now a horizontal line. The line c. 400, c. 700, c. 1000 and c. 1300 m respectively. shown in Figure 3 corresponds to z = 2 km at Thus the penetration distances are [(y-x)=] Pe=0.98MPa; equation (7) shows that the about 100, 300, 500 and 700 m respectively. At a corresponding values of Pe for z= l km and magma rise speed of 1 m s -l, the corresponding 3 km are 0.49 and 1.47 MPa, respectively. Thus, dyke emplacement times would range from for all cases where the dyke tip penetrates into about 100 to 700s and in these time intervals and stalls within the ice, the excess pressure in any temperature changes caused solely by ther- the tip can be positive but less than about 0.5- mal conduction would penetrate a distance (d) of 1.5MPa, the exact value depending on z. The order (nt) 1/2 where n is the thermal diffusivity of requirement that Pt be no less than c. 3.3 MPa the ice or chilling dyke magma. Thermal diffu- leads to the truncation of the lines in Figure 3, sivities of both ice and basalt are c. 10 -6 m 2 s -1 and so the excess pressure Pc can also become and so d would be at most a few centimetres. negative by up to about -10 MPa. Thus englacial dykes could well be emplaced in None of the dykes modelled above (using the initial phase of an eruption (Fig. 1). plausible magma densities and volatile contents) Soon after their emplacement, dykes intruded are expected to break through to the upper into ice would provide relatively efficient ice surface of the ice. Thus magmatic eruptions at melting because of the formation of two broad the surfaces of glaciers and ice-caps should not and extensive surface areas (the sides of the be a common occurrence even when dykes do dyke) in contact with the ice. Anticipating penetrate into overlying ice. However, there calculations given below for heat loss from a are some potential consequences of the fact sill, typical average heat transfer rates during the that the pressure differential, Pe, between the first 10 seconds after emplacement exceed water vapour in a dyke tip and the surround- 3 MWm -2, and this could be a factor in the ing ice could range from c. 1.5 MPa positive to rapid initial production of meltwater reported in c. 10MPa negative. Positive pressure differen- some Icelandic eruptions (Gudmundsson et al. tials this small will probably not lead to brittle 1997). Over the subsequent few tens of hours, failure of the surrounding ice, being less than the solidification of the magma and formation of likely tensile strength of the ice, but large cooling cracks, together with melting of adjacent negative pressure differentials may lead to fail- ice, would almost certainly cause the magma ure in tension or shear of the ice forming the column to lose coherence and collapse to form a dyke walls and collapse of blocks of ice into 'dyke rubble pile'. If the dyke were c. 200 m high the gas cavity at the dyke tip. This process would and c. 1 metre wide (200 m 2 cross-sectional area), be encouraged by the c. 8% volume decrease then its eventual collapse could produce a rubble which occurs when ice melts to water. Progres- pile at least c. 15 m wide by 15 m high even with sive collapse might occur until a pressure path to minimal bulking (or more likely c. 20 m wide by the surface was formed, in which case the excess c. 10 m high if it eventually attained angle of rest water vapour pressure in the dyke tip would be slopes). The cores of eruptive structures begin- vented to the atmosphere and the consequent ning with this type of event might contain a unloading of the magma would lead to further breccia pile with morphology diagnostic of its magma vesiculation and the onset of explosive dyke-induced origin. activity. This activity would almost certainly be We now turn our attention to the conse- phreatomagmatic because of the intimate con- quences of magma intruding at the ice-rock tact between magma, water and spalled blocks interface instead of propagating as a dyke into of ice. It would not be long-lived, however: even the ice. complete relaxation of the pressure at the top of the magma column to atmospheric pressure would not cause magma to rise to the surface Sill intrusion at the ice-basalt substrate of the ice, and so the magma at the top of the interface column would rapidly be chilled, causing explosive activity to cease. The conditions that determine where the tip of We can obtain an idea of the timescale for an initially vertically propagating dyke ceases to the dyke emplacement process using the typical move upwards, and instead initiates a fracture magma rise speed, c. l ms -~, quoted earlier. propagating sideways to allow the intrusion of a Figure 1 shows that dykes will penetrate a sill, are complex. Lister (1990), in modelling the distance (y-x) into the ice layer. Figure 2 shows rise of mafic magmas from deep levels, has that, as y takes the values 500, 1000, 1500 and argued that lateral intrusion will be favoured at 2000 m, the value of x at which Pt is equal to the the level of magma neutral buoyancy, and this is Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS l l

ice

Fig. 4. Successive stages in the intrusion of a sill at the base of an ice layer. The thickness of the sill relative to its horizontal extent is exaggerated for clarity. an attractive model for the origin of crustal the bulk magma density averaged over the verti- magma reservoirs (e.g. Ryan 1987). However, at cal extent of the feeder dyke would lie within shallower levels in general, and especially when e. 10% of 2250kgm -3. Thus if Pr equals the tip of a dyke is nearing the shallowest level 2300 kg m -3, the value of(pr -/3) will lie between to which the stresses controlling it will allow it to about +75 and - 175 kg m -3. Then since Pi equals penetrate, it seems inevitable that local varia- 917kg m-3, Pc is dominated by the first term in tions in host rock properties will also play a equation (8) and is only a little greater than the part. We infer that dykes capable of penetrating weight of the overlying lithostatic (cryostatic) a significant distance into overlying ice will not load. Thus as the sill grows, Pi increases from Pt be excessively sensitive to the presence of the towards Pc, just attaining this value when sill large density contrast at the rock-ice interface, growth ceases and the pressure gradients due to whereas those which would otherwise have magma motion vanish. Furthermore, Rubin stalled just above the interface will initiate sill (1993) showed that most of the pressure decrease intrusion even if part of the magma rises into a used to overcome friction will occur over a dyke somewhat overshooting the interface. disproportionately short distance near the dyke When magma rises in a dyke and then intrudes tip. As a result, the pressure in nearly all of the sill as a sill, there must be a finite vertical pressure will be quite close to Pc for most of the duration gradient in the sub-vertical feeder dyke due to the of its emplacement after the brief initial period weight of the magma; if the sill is intruded hori- when most of the sill consists of 'tip'. zontally, there is of course no pressure gradient We use this fact in Table 1 to illustrate in the sill due to magma weight. However, conditions in a typical mafic magma intruded at a pressure gradient required to overcome wall the interface between glaciers of various thick- friction associated with magma flow must exist nesses, y and the underlying silicate surface. The in both the vertical and the horizontal parts of magma reservoir depth is assumed to be just the system. Figure 4 shows the geometry at greater than 1 km so that [g z (pr -/3)] = 0.5 MPa, various stages during the sill injection. The and so the sill pressure Pc exceeds the overlying injection pressure Pi in the magma at the ice- load by this amount. At great depths the magma rock interface is equal to Pt at the moment sill contains 0.25 mass% H20 and 0.2 mass% COz injection begins and increases thereafter, but as before; the table shows the amounts of these must always be less (because of the pressure volatile phases exsolved, the bulk magma den- difference required to drive magma motion sity, and the volume proportion of the magma against wall friction losses) than the pressure in consisting of gas bubbles for a sill intruded under a static column of magma extending from the various ice thicknesses (y) from 50 m to 2000 m. reservoir up to this point, which Figure 1 shows The entry in Table 1 for y--303 m corresponds to be Pc given by to a sill pressure of 3.3 MPa, which is the water saturation pressure for the assumed water con- Pc --- Pa +gYPi +gzpr -gz/3 tent of 0.25 mass% . The fact that under = Pa W gy pi + gZ(pr -- /3) (8) shallower ice thicknesses the sill inlet pressure must be less than this value inevitably implies where Pa is the atmospheric pressure, c. 0.1 MPa. that excessive amounts of water vapour would Earlier we specified that y would probably lie in have to be exsolved in the dyke tip during such the range 500 to 2000m and that z would lie intrusions. Indeed, it calls into question the in the range 1000 to 3000 m. Also, we found that advisability of ever assuming that the pressure in Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

12 L. WILSON & J. W. HEAD

Table 1. Illustration of the amounts of CO 2 and H20 exsolved from a mafic magma intruding beneath glacial ice layers of various thicknesses and the consequences for the bulk density of the magma and the volume fraction of the magma that consists of gas bubbles

Glacial ice Pressure in Exsolved Exsolved Magma bulk Exsolved gas thickness (m) most of sill CO2 amount H20 amount density proportion (MPa) (mass%) (mass%) (kg/m 3) (volume%)

50 1.049 0.19903 0.13856 557 79.4 100 1.499 0.19876 0.10692 817 69.8 250 2.847 0.19795 0.02583 1600 40.9 303 3.327 0.19766 0.0 1869 30.9 500 5.093 0.19660 0.0 2096 22.5 1000 9.587 0.19391 0.0 2348 13.2 1500 14.080 0.19121 0.0 2454 9.3 2000 18.573 0.18852 0.0 2513 7.1

The total volatile content of the magma prior to any gas exsolution is 0.2 mass% CO2 and 0.25 mass% H20. See text for discussion.

the tip of a propagating dyke is exactly buffered become more concentrated at the top of the sill, by the saturation pressure of the most soluble where bubbles will eventually burst to release magma volatile, especially if that volatile is gas into a continuous pocket. Formation of a present in large amounts. For the case shown in continuous gas phase facilitates gas loss through Table 1 any magma containing more than any fractures which exist or subsequently form 0.25 mass% water would exsolve a significant in the overlying ice. However, the timescale for amount of that water if it were intruded under an gas loss will be determined by the rise speed ice thickness less than 300 m. For ice thicknesses of the gas bubbles and the thickness of the sill. less than about 100 m the gas volume fraction in A typical timescale can be illustrated by con- much of the sill would be greater than 70%, and sidering the sill intruded under 303 m of ice in spontaneous magma fragmentation would be Table 1. With the assumed carbon dioxide con- expected. Enhanced interaction between magma tent of 0.2 mass% gas bubbles will have clots, rapidly chilled by intimate contact with the nucleated at a pressure of 338 MPa, deep in the water being produced, would lead to the forma- magma source region, and they are expected to tion of hyaloclastite breccia. It seems quite have initial diameters of c. 20 microns (Sparks possible that, under these circumstances, sill 1978). By the time that they have decompressed formation would not occur; instead, continuing to the sill pressure of 3.3 MPa they will have explosive instability would lead to the formation expanded to diameters of c. 93 microns (assum- of a hyaloclastite ridge, or series of cones, along ing that the emplacement time will have been the feeder dyke. short enough that little diffusion of gas into the The ice thickness under which magma disrup- bubbles occurs; Sparks 1978). The bubbles will tion of this kind will occur will be a function of by this stage be rising through the magma at a the magma water content. Table 2 shows the speed u determined by the balance between their water saturation pressures corresponding to a range of magma water contents and the minimum ice thicknesses needed to suppress magma Table 2. Examples of the minimum ice thicknesses needed disruption during intrusion. We stress that these to suppress spontaneous magma disruption during sill are only approximate depths, because an exact injection as a function of the magma water content calculation would have to take more detailed Magma Water Required account of the pressure distribution and density water saturation minimum structure of the magma throughout the dyke-sill content pressure ice thickness system. However, this important influence of ice (mass%) (MPa) (m) thickness may explain some of the differences observed between the subglacial deposits of local 1.0 24.0 563 or alpine glaciers and continental-scale glaciers 0.5 9.0 245 (Smellie & Skilling 1994). 0.25 3.3 98 0.125 1.2 28 In all of the cases shown in Table 1 which intrude as undisrupted magmatic liquids, the gas The magma is assumed to contain 0.2 mass% CO2 in bubbles will drift upward through the liquid to addition to the amount of H20 shown. Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 13 buoyancy (given by [4/317rr3[/3 - ~rg]g where r is still thin, there may be the possibility of minor the bubble radius, /3 is the magma density, O-g instabilities causing the front of the sill to grow is the gas density and g is the acceleration due to initially as a series of pahoehoe-like toes; as the gravity) and the drag force acting on them sill extends and thickens, however, we expect any (61tr/mrU where r/m is the magma viscosity). Using such toes to be overidden by the more nearly

/3 = 2250 kgm -3, g=9.8ms -2 and Tim = 100Pa s sheet-like intrusion. for a mafic magma, and neglecting O-g because it To quantify some of these considerations, is approximately 100 times smaller than/3, the consider a basaltic shield volcano having a rise speed of a 93 micron diameter carbon magma reservoir within which an excess pres- dioxide bubble is 0.42/ares -J At this speed, sure of 1 MPa causes a dyke to propagate to the about four weeks are needed to segregate all of surface. The length of the dyke, A, is equal to the bubbles from a sill one metre thick; pro- the depth of the roof of the reservoir, say 2 km portionally greater times are needed for thicker (e.g. Gudmundsson 1987; Ryan 1987). The mean sills, and sills intruded under thicker ice layers width of the dyke, W, will be given by will have smaller bubbles with longer rise times. These timescales are much longer than the W--[(1 - v/~t](rt/2)PA, (9) emplacement times of the sills (at most a few where v and la are the Poisson's ratio and shear hours given the typical rise speeds of mafic mag- modulus for the crustal rocks, c. 0.25 and 3 GPa, mas) and so gas loss can be ignored in all cases. respectively (Rubin 1993), so that W equals Magma intruding into a sill spreads sideways 0.8m. The excess pressure drives magma with and, if the ice-rock interface is inclined, prefer- viscosity Tim upward through the dyke at an entially downslope. Magma in a sill probably average speed UM where, if the magma motion is always forms a thinner and more widespread laminar, layer than lava in a surface flow with the same mass flux and hence causes a more geometrically UM = (W2e)/(8~mA). (lO) efficient transfer of heat to the ice; this may be a second explanation for the initial abundance of If r/m=100Pas, UM=C. 0.4ms -1. The Rey- meltwater that is observed in some Icelandic nolds number for the magma motion is subglacial eruptions (Gudmundsson et al. 1997). We base this assertion on the following series of Re = (2WUM/3)/~m, (11) arguments. The thicknesses of subaerial flows are where/3 is the magma density, c. 2200 kg m -3, in determined by the bulk density, viscosity and which case Re = c. 14, confirming that the mag- effective yield strength of the magma, the accel- ma motion is laminar. The total volume flux, eration due to gravity, and the surface slope: the V through the dyke is the product of the magma requirements are that in the levees the stress at speed UM, the dyke width W and the horizontal the base (the product of levee thickness, gravity extent of the dyke, L. Assuming that L is of the and ground slope) is equal to the effective magma same order as A, say 1 km, we find V equals yield strength, and that in the central channels 320m 3 s -1. For comparison, this value is quite the product of channel width, magma depth and similar to the c. 200 m 3 s-1 eruption rates typical magma flow speed (the speed being in turn con- of recent basaltic activity on the East Rift trolled by flow depth, magma viscocity and Zone of Kilauea volcano, Hawai'i (Wolfe et al. ground slope) must equal the volume flux from 1987; Parfitt & Wilson 1994). the vent (e.g. Pinkerton & Wilson 1994). The Assume first that this dyke feeds a subaerial same is true for pahoehoe toes with the com- basaltic lava flow which has a thickness D, plication that a yield-strength-like component of density p, effective yield strength Y, viscosity ~L, the magma rheology, in addition to the other and is moving down a slope o~. Heslop et al. factors, influences the 'central channel' depth. (1989) analysed the fluid mechanics of the In contrast to this, a subglacial flow or sill has no proximal parts of a flow on the south edge of free upper surface. The thicknesses of the 'levees' the summit caldera of Kilauea volcano for and the 'central channel' are controlled only by which typically D is c. 2m, p is c. 1000kgm -3, the stress distribution in the host rocks. On the Yis c. 700 Pa, r/L is C. 50 Pa s and c~ is 2 °. The mean largest spatial scales, that same stress distribu- advance speed, Ue, of such a flow is given by tion prevents the sill from thickening locally into a series of lava flow-like fingers in the same way UL = (pgsin o~D2)/(3~L), (12) that vertically propagating dykes travel upward as sheets of finite lateral extent, not as a series of so that in this case UL is c. 9ms -l. To accom- nearby tubes. On smaller spatial scales, especially modate the total estimated magma volume flux in the early stages of growth of a sill while it is of V=320m3s -1, the width of the flow must Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

14 L. WILSON & J. W. HEAD

d w

II Fig. 5. Details of the development of a chilled crust, thickness dc, and an overlying meltwater lens, thickness dw, during the progressive intrusion of a subglacial sill of thickness ds. then be about 18 m, in good agreement with the from which ds can be found as a function of observed width. time by substituting equation (15) for E in equa- Now assume that, instead of erupting, the tion (14): dyke magma ceases to propagate upward when it encounters the base of an ice layer 500 m thick d~ = {[(1 - v)/~t](K/Z)ps}l/Z(u/L)l/2t 1/2. (16) (so that the magma density is close to Finally, from Us = (dE/dt) we have 2100kgm -3, see Table 1) and intrudes as a horizontal sill. Initially the sill will extend along Us = 0.511//{[(1 - v)/~tl(x/Z)PsL}ll/2t-1/2. (17) the entire 1 km horizontal extent of the dyke (L) Still using V equal to 320m 3 s -l and L equal and will be growing laterally away from it on to 1 km, and taking Ps as 5 MPa, a suitable value both sides (Figs 4 & 5). Let the proximal sill for a sill intruded under about 500 m of ice (see thickness be ds (Fig. 5) and the magma flow Table 1), we find the values of E, d~ and Us as a speed be Us. The total volume flux must be the function of time shown in Table 3. After the first same as that in the dyke and so d~ and Us are second the sill has grown horizontally to 13 m on related via

V= 2dsLUs. (13) Table 3. Variation with time, t of the extent, E, However, the sill grows by deforming the host thickness, ds, and magma inflow speed, Us, for a sill materials (rock below, ice above) in an elastic driven by an injection pressure of Ps = 5 MPa from a c. 0.8 m thick dyke I km in horizontal extent when the manner, and the elastic properties of ice are not volume flux (V) is 320 m 3 s-I grossly different from those of rock (Hobbs 1974). Let the horizontal extent of the sill on t E d~ Us either side of the dyke be E at time t and let the (s) (m) (m) (ms -I ) magma pressure at the point of injection be Ps. Then by analogy with equation (9), 1 13 0.025 6.38 3 22 0.043 3.69 ds = [(1 - v)/~t](n/2)PsE. (14) 10 40 0.079 2.02 By eliminating ds between equations (13) and (14) 30 70 0.14 1.17 and noting that by definition Us equals (dE/dt), 100 128 0.25 0.64 300 221 0.43 0.37 we find the relationship V equals 2[(1 - v)/g] × 1000 404 0.79 0.20 (n/2)P~EL(dE/dt), which integrates to give E as a 3 000 699 1.37 0.12 function of time: 10 000 1277 2.51 0.064 E= iV/{[(1 - v)/g](rt/2)PsL}]'/2t 1/2, (15) 30000 2211 4.34 0.037 Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 15 either side of the dyke; its advance speed of the ice. Their analysis also effectively assumed c. 6ms -1 is already smaller than the c. 9ms -1 that the overlying ice and underlying rock behave advance rate of the surface flow described above. in a rigid fashion. The fact that water is denser Although the sill is only c. 25 mm thick, its than ice, leading to a volume decrease on melt- surface area in contact with overlying ice by this ing, potentially provides some of the volume time is 26 000 m 2, whereas the surface area of the needed to accommodate the magma. Addition- 2m thick, 18m wide lava flow after it has ally, if the pressure in the water increases, some advanced 13 m is only 234m 2. Only if the sill magma volume is accommodated by the small grows to a distance of 6.4 km from the dyke will but finite amount of compression of the water it have a mean thickness as large as the 2 m thick- produced (water is much more compressible ness of the lava flow; the magma injection speed than the overlying ice, the magma, or the under- of the sill will then be c. 0.1 m s- 1, two orders of lying rock). If the water pressure becomes large magnitude less than the lava flow advance rate. enough to support the weight of the entire over- Note that, even as early as one second after lying ice layer, then sudden and large-scale (but the start of sill injection, the thickness of the short-lived) escape of the water along the margins chilled skin on the sill magma, c. (t~mt)1/2 equals of the ice-rock contact becomes possible. c. 1 mm, where ~m is the thermal diffusivity of We are not convinced that this is how the basalt, is much less than the c. 25 mm thickness system behaves. The injection of magma into a of the sill, so heat transfer to the overlying ice sill fed by a dyke explicitly requires some defor- does not hinder sill injection in this example. mation and local compression of the adjacent Admittedly, the above comparison has var- host materials as typified, for example, by the ious deficiencies. For example, by the time the shear modulus and Poisson's ratio in equation sill extends horizontally for a distance compar- (9). The fact that the host material overlying the able to the along-strike length of the dyke sill is ice rather than rock does not change this. (c. 1 km in the earlier examples), it will be Any water created by ice melting is a Newtonian spreading sideways, i.e. its horizontal growth fluid and transmits stresses isotropically (as does will be taking place parallel to the strike of the the unchilled part of the magma as long as its dyke as well as normal to it, and by the time it properties are near-Newtonian), so it is not has extended to approximately twice this dis- appropriate to consider pressure changes in the tance magma will be flowing more nearly water independently of the pressure in the rest radially away from the source region, and so of the fluid system. Indeed, any potential the continuity relationship used above will pressure increase in the water (possibly caused, overestimate the sill thickness and advance for example, by the very rapid conversion of a rate. Also, it has been tacitly assumed that the thin film of ice directly to supercritical vapour at excess pressure in the magma is preserved the magma-ice contact) would first be accom- throughout the emplacement event, whereas in modated by the compression of the bubbles of fact that pressure is likely to decrease steadily as exsolved carbon dioxide in the adjacent magma. the magma reservoir at depth is deflated by In our view, the melting of ice into water during magma removal. Further, the stress distribution the intrusion process, and the consequent reduc- in the feeder dyke has been assumed to remain tion in volume of the H20 component (due to constant, whereas in fact the growth of the sill liquid water being denser than ice), simply makes will have a feedback effect on the overall it possible to inject a greater volume of magma geometry of the dyke-sill system, changing the for a given set of magma pressure conditions. magma volume flux to some extent. Even so, the We do, however, agree with the analysis of comparison serves to support the assertion that H6skuldsson & Sparks (1997) as regards the rate sills generally have a larger contact area with of cooling of the injected magma and melting of adjacent ice than equivalent surface lava flows. the overlying ice, and now develop these ideas to Some of the consequences of the injection of illustrate the importance of the magma injection magma beneath an ice layer were investigated by rate and the ultimate consequences of the H6skuldsson & Sparks (1997), who evaluated the intrusion process. Figure 5 shows schematically variations with time of the thickness of the chilled the thickening of the sill, its chilled crust and the crust on the magma and also the heat loss rate overlying water layer, and defines the total thick- from the magma, and hence the thickness of ice ness of the sill, ds, and the sill crust thickness, arc, melted. They did not, however, deal explicitly near the sill injection point. The corresponding with the rate of thickening of the magma layer, depth of ice melted is di. Using treatments based instead introducing an efficiency factor which on those developed by Carslaw & Jaeger (1947), represented the fraction of the heat available H6skuldsson & Sparks (1997) give the crust from the magma that was actually transferred to thickness, do, and the heat loss rate per unit Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

16 L. WILSON & J. W. HEAD area of magmaice contact, q, as a function of the increase in the crust thickness with time time, t, as therefore imply a minimum magma injection rate into the sill. In the example of sill injec- dc = 2A(t~m t) 1/2, (18) tion calculated earlier for comparison with an equivalent volume-flux lava flow, we saw that q = [kin(Tin - Tw)]/[erf(A)Ot~mt)~/2], (19) the sill was easily able to avoid excess cooling. To establish the minimum magma volume flux where Tm is the temperature of the uncooled sill to allow sill injection to be thermally viable, we magma, Tw is the temperature of the meltwater note that the essential requirement is that the above the crust, ~m and km are the thermal sill thickness ds given by equation (16) must diffusivity and thermal conductivity, respec- exceed the chilled crust thickness dc given by tively, of the solidified magma, and A is a equation (18). Both have the same time depen- constant given by the solution of dence, and so the requirement is simply [A-1 exp( - A2)]/[erf(A)] {[(1- v)/I,t](n/Z)ps}l/2(V/L) 1/2 >> 2An~ 2 (23) = [rcl/2Lm]/[cm(Tm - Tw)] (20) which, since A equals 1.1514, is more conveni- where Lm and cm are the latent heat of fusion ently written and the specific heat, respectively, of solidified magma. Taking Lm as 2.09 x 105 J kg -1, Cm as [(1 - v)/~t](rc/Z)Ps(V/L) >> 5.3rim (24) 1200Jkg -1K -t, Tm as 1473K (1200°C), and Tw as 277 K (i.e. close to the melting point and just We saw in Table 1 that Ps probably lies be- above the temperature at which water has its tween 3 and 18MPa; nm is c. 0.8 x 10-6mZs -l, maximum density) we find A equals 1.1514 and and we have v = c. 0.25 and ~t = c. 3 GPa. Thus eft(A) equals 0.8968. We note that H6skuldsson the requirement is essentially that (V/L) should & Sparks (1997) found erf(k) = 0.84, and suspect be greater than a critical value which lies that they inadvertently used the latent heat of between 6x 10 -4 and 36 x 10-4mZs -1. Some fusion of ice, rather than that of magma, in solv- values of (V/L) observed in, or deduced for, ing equation (20), but this does not lead to any subaerial eruptions include c. 3m 2s -1 for the major differences between their results and ours. 1961 fissure eruption at Askja, Iceland (Thorar- We now integrate equation (19) to find the total insson & Sigvaldason 1962), c. 0.6m 2 s -~ for the amount of heat absorbed by the ice and the 1783 Lakagigar eruption in Iceland (Thorarins- resulting water as a function of time, H(t): son 1969), c. 7 m 2 s -1 for the July, 1974 summit eruption of Kilauea, Hawai'i (Heslop et al. 1989) and 12m2s -1 for the Yakima member of the H(t) = q(t') dt' Columbia River Basalt series (Swanson et al. 1975). These are all orders of magnitude greater : {[2km(Tm - Tw)l/erf(A)}[t/(rtr;m)] 1/2 (21) than the minimum flux required, and so it seems likely that sill injection beneath ice should be a and equate this to the amount of heat needed to common occurrence, uninhibited by cooling melt the thickness di of ice, di = HI(piLl), where problems, when the stress regime favours it. pi and Li are the density and latent heat of fusion of ice, respectively, giving

di = {[2km(Tm- Tw)]/[piLierf(A)]} Further stages of activity

× [t/(~l~m)] 1/2 (22) When cooling does not limit sill injection at an ice-rock interface, magma injection will con- Using these results, the first five columns of tinue until one of two possible events happens: Table 4 show how q, H, di and dc are expected to (1) the supply of magma from the source feeding change with time. Also shown is the thickness of the eruption ceases because the stresses driving the water layer produced by the ice melting, dw, the magma have been relaxed; (2) the sill spreads given by dw = di(pi/pw) = c. 0.917di. far enough laterally that the stresses at the It has tacitly been assumed in the above propagating tip of the sill cause the precursor analysis that the sill is injected fast enough that fracture (recall that the tip of the sill will contain the total sill thickness at the vent, ds, is greater pressurized water vapour and not magma) to than the chilled crust thickness de; in other reach the edge of the ice pile so that a connection words, there is some uncooled magma in the is made to the atmosphere. We now consider the core of the sill. The calculations given above for consequences of these events in turn. Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 17

Magma supply ceases net uplift of the ice mass, subsidence of the overlying ice ultimately occurs. There seems no If the magma supply is cut off, growth of the sill reason why any activity should be seen at the immediately ceases, but melting of overlying ice surface other than slow ice subsidence, greatest continues as heat is convected upward through over the vent, to form an ice cauldron (Fig. 6). the water layer in contact with the sill. Because Only if the accumulated water escapes, either of the volume reduction as ice melts to water, the by slow seepage or by sudden release in a first consequence of this is that any residual j6kulhlaup, will there be more complex short excess pressure which may be present in the term topographical changes (Bj6rnsson 1992). magma, together with any residual non-hydro- Of course, if water does not escape, it will static vertical compressive stress which may be eventually freeze again, and its expansion as this present in the overlying ice, is quickly relaxed. happens will induce new stresses in the ice layer. Subsequently, if the overlying ice cannot deform However, the freezing process will be so slow (the downward fast enough, or alternatively if expan- timescale for conductive heat loss from under sion of gas bubbles in the as-yet unsolidified part 100 m of ice is about 300 years) that any required of the sill magma cannot crack the cooled crust ice deformation will probably be by plastic creep. and expand the sill sufficiently, a gap filled with Finally, we note that the heat sharing calcu- water vapour will exist between the water and the lation just employed assumes that all of the heat overlying ice. Assuming that the temperature in lost from the sill causes ice melting. This may not the convecting water remains at only a few K be the case. The temperature of the water be- above the melting point, the absolute pressure in tween the sill and the ice is by definition higher the water would have decreased to c. 103 Pa, than the ice melting point, whereas the ice itself i.e. 10 -2 atmospheres, by the time that a vapour must have a temperature at or below the melting phase appeared. An absolute pressure this low point. If the ice temperature is even infinitesi- would cause an enormous stress gradient in the mally below the melting point, some heat is overlying ice and an equally impressive pressure conducted into the ice ahead of the melting front gradient across the chilled margin of the under- and is not available to supply latent heat to melt lying sill, and so probably in practice no vapour ice. However, this is not a large effect. Consider layer ever forms. However, if it did so it would the ds = 4.3 m thick sill intruded on a timescale of form a good insulator: the vapour density would 3 × 104s=c. 8 hours illustrated in Table 3. be c. 10-2kgm -3 and so, although the specific According to the above calculation this sill heat of the vapour is only a factor of approxi- could generate a 4.3 × 13.3 = c. 57 m deep water mately two smaller than that of liquid water, the layer. The timescale for cooling the sill is thermal capacity per unit volume of the vapour c. [d2/t~m] = c. 1.8 × 107 s = c. 200 days. On this would be c. 2 x 105 times smaller. Presumably in time scale a thermal wave would penetrate a practice an equilibrium will be reached between comparable c. 4m distance into the ice ahead ice deformation, sill inflation, ice melting and of the melting front. Assume that the ice was heat transfer in which an appropriately narrow as much as 10 K below the freezing point. vapour space exists (if it exists at all). Then the average amount of ice heating would This process will continue until the available be c. 5 K and the amount of heat leaked into sill magma heat content is exhausted. An upper the ice per unit area would be c. 4 m x 5K × limit on the thickness of ice which can be melted 2100Jkg -l K -1 x 917kgm -3 = 3.9 x 107jm -2. by a given thickness of magma can be found by The amount of heat contained in the c. 57m assuming that heat transfer through water and thick layer of water (heated to 4K above water vapour continues to cause ice melting the melting point) would be c. 57 m x4K x until all of the magma has cooled to 274 K, the 4200Jkg -1K -1 x 1000kgm -3 =9.6 x 108jm -2. temperature at which water has its maximum This suggests that the heat transfer to the water density, at which point convective heat trans- is more than 95% efficient. In contrast, the heat- fer ceases. On this basis, and assuming no net sharing calculation employed earlier shows that lateral transfer of heat, each one metre thickness the efficiency of the process would have to be of sill magma could melt (pm[Lm q- cm(Tm - Tw)]/ less than 83% before there was no net sub- (piLi)= 14.5 m thickness of ice, to form a water sidence of the ice. layer 14.5 x (pi/pw)~ 13.3 metres deep. The There is a potentially useful diagnostic con- (14.5- 13.3 =) 1.2 metres of space thus created sequence of activity in which the intruded sill is by the time ice melting ceases more than never exposed to atmospheric pressure. With ice accommodates the one metre thickness of overburdens of several hundred metres, basaltic magma intruded, and so although the initial magmas should typically exsolve most of their intrusion of the sill must have caused some small CO2 but little of their H20. Thus, as pointed out Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

18 L. WILSON & J. W. HEAD

(a) (b)

Fig. 6. Successive events during and after the intrusion of a sill at the base of an ice layer when the sill does not reach the edge of the ice sheet. (a) Early stage of intrusion; (b) sill has grown in all directions, chilled crust and overlying water lens are both thicker; (c) sill growth has ceased due to termination of magma supply, chilled crust and water lens have both thickened, and some subsidence of the surface of the ice has begun because the ice-to-water volume decrease has more than compensated for the sill thickness; (d) all available heat has been extracted from sill and vertical extents of water lens and subsidence have reached their maximum values. by Dixon et al. (2002), analysis of the residual to the hydrostatic weight of the overlying ice; we CO2 and H20 contents of eruption products showed earlier that the pressure in the sill is should help distinguish between magma that has always fairly close to this value, so no major been emplaced under an ice overburden and that change in the overall magma flow rate through which has been erupted subaerially. and into the sill will occur at this stage. However, as soon as a significant amount of water has drained from above the sill, the A pathway to the edge of the ice forms pressure in this region will start to decrease toward atmospheric pressure, because the water As soon as a growing sill (Fig. 7a) approaches can be replaced by atmospheric air leaking in. close enough to the edge of the ice cover that a Only if the overlying ice can deform on a short direct connection between the intruded materials enough timescale to replace the water will ice and the atmosphere is made (Fig. 7b), the automatically stay in close proximity to the top pressure in the sill tip will decrease to that of of the sill magma. For a set of conditions similar the atmosphere as the pressurized water vapour to that envisaged here, H6skuldsson & Sparks escapes. The elastic constraints on the aspect (1997) calculated an ice deformation rate of ratio of the sill will then decay very quickly as the order 1 mm s-l, so if the rate of thinning of the water which has already been produced above water layer exceeds this value, the pressure will the sill begins to leak out onto the surrounding inevitably start to decrease. surface. For a short time, the pressure acting at Any pressure reduction in the sill will lead to the magma-water interface will become equal an increase in the pressure difference between the Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 19 (a) (b) chilled crust water water ( . (' ~ce sheet~ ~..r.7.../z~k~_) dyke---~[ I' sill magma II v~atercStpIles

explosivesill (c) (d) fragmentation

as water drains out

(e) (f) collapsed ice

fountain II

Fig. 7. Successive events during and after the intrusion of a sill at the base of an ice layer when the sill extends as far as the edge of the ice sheet. (a) Early stage of intrusion; (b) sill has reached edge of ice sheet and some water leakage begins; (c) much of the water generated earlier has drained out from beneath the ice and reduced interface pressure has allowed additional magma vesiculation; (d) pressure has become low enough near the drainage point for sill magma fragmentation to begin, enhancing the heat transfer to overlying ice; (e) all of the sill has been disrupted and a lava fountain exists at the outlet of the feeder dyke, rapidly eroding the overlying ice and feeding a subglacial lava flow; (f) the equivalent of stage (e) when the overlying ice has collapsed, greatly increasing the efficiency of thermal contact between the lava flow and ice. bottom and the top of the dyke, and hence an in the water is greatest near the dyke and least at increase, albeit probably small, in the magma the edge of the ice sheet, the resulting pressure flow rate through the dyke system. It may also gradient driving the water toward the exit. How- have dramatic consequences, because it will lead ever, this state of affairs cannot persist for long. to gas exsolution from the sill magma beneath It seems inevitable that, as water drainage the chilled crust. In the initial stages, the magma becomes more efficient, and the flowing water will simply vesiculate: existing carbon dioxide itself begins to melt and erode overlying ice, the bubbles will expand and new bubbles of both pressure at the sill-ice contact will decrease to CO2 and H20 will form at a rate which causes the approach the atmospheric pressure. The key magma to stay in physical contact with the issue is then whether or not the magma contains overlying ice (Fig. 7c). This will lead to continued enough volatiles so that at atmospheric pressure water production, and the system will tend the volume fraction of gas bubbles in the magma toward a new equilibrium in which the pressure becomes so great that magma fragmentation Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

20 L. WILSON & J. W. HEAD begins to occur. For the plausible volatile mix- not clear what fraction of the fragmented ture used earlier (0.2 mass% CO2, 0.25 mass% magma would be washed out with the escaping H20), magma fragmentation would begin at water and what fraction would be left behind to about 1.2MPa, i.e. 12bars, and so we assume form a hyaloclastite deposit. that such fragmentation is common. A major change occurs when the wave of Since the lowest pressure in the system must magma disruption reaches the feeder dyke. always be at the distal end of the sill closest During the fragmentation process magma is to the connection to the atmosphere, it is in still flowing up the dyke and being injected into this region that magma fragmentation will begin. the sill. However, the fragmentation process As the pressure in the space above the chilled greatly reduces the frictional energy losses magma crust decreases, the crust will initially associated with magma motion in the sill and prevent any response from the underlying so as soon as all of the sill magma has been magma. However, due to the presence of cooling fragmented, the flow rate up the dyke will cracks in its outermost parts, the crust is unlikely inevitably increase somewhat. The pressure at to have great strength. Once the pressure differ- the dyke outlet will now be very close to ence across the crust exceeds this strength it atmospheric, and so the system will behave just will fail, and an expansion wave will propa- as it would have done if the eruption had started gate vertically downward into the sill. The subaerially. A chain of lava fountains will form speed of the wave will be some fraction of along the dyke and magma clots falling from the the speed of sound in the vesicular magma, at fountains will begin to form lava flows (Head & most c. 100ms -l (Kieffer 1977; Wilson & Head Wilson 1989). The lava fountains will impinge 1981). Thus for a sill a few metres thick (Table 3) on the overlying ice, greatly increasing the ice the timescale will be only a few hundredths of a melting rate above the dyke (Fig. 7e). The second. Passage of the expansion wave will resulting cavity 'drilled' into the overlying ice fragment the magma, and expansion of the will grow upward until the subaerial height of released gas through a pressure difference equal the lava fountain is reached (Head & Wilson to the effective crustal strength will accelerate 1987), after which heat will only be transferred disrupted magma clots to impact the overlying to the ice by radiation from pyroclasts in the ice (Fig. 7d). As an illustration, formulae given fountain. From this time onward a new balance by Wilson (1980) for transient explosions show between ice subsidence and melting will be that if the effective strength were 1 MPa, then established but, if the eruption continues for under a 500m thick ice layer where the sill long enough, it is clear that the explosive activity pressure was c. 5 MPa (see Table 1), expansion may eventually emerge through the ice; interac- of the c. 0.2 mass% of CO2 from c. 5 MPa to tion with the water being produced will cause the c. 4 MPa would generate speeds in the hot vesicu- activity observed to be phreatomagmatic. This lated pyroclasts up to c. 30ms -j. This should scenario would be complicated somewhat if the result in locally enhanced ice melting and magma ice layer above the now fragmented sill residue chilling, and might be enough to trigger a underwent fracturing and collapse rather than sustained violent fuel-coolant type of interaction slow plastic deformation (Fig. 7f). In this case (Wohletz & McQueen 1984; Zimanowski et al. the pressure acting at the exit from the dyke 1991). The products of the explosive mixing would still be very close to atmospheric as long would be directed toward the exit to the atmo- as there was a reasonably high porosity and sphere, and the wave of pressure reduction, permeability in the collapsed ice block pile, but vesiculation and fragmentation would also pro- the interaction between the magma and the ice pagate from the distal end of the sill toward the would be more vigorous because of the tendency feeder dyke. In this case the propagation speed of ice blocks to settle as their bases were melted. would be a balance between the speed of the There is a second possible consequence of wave front into the unaffected sill magma (again efficient water drainage once a pressure pathway some fraction of the local speed of sound) to the atmosphere is established, one which is and the speed at which water and fragmented particularly applicable to magmas that do not magma could be expelled from the discharge have a large volatile content. As soon as the region. The rate of escape will be influenced elastic constraint on the shape of the magma-ice mainly by the lateral extent of the sill; we saw contact is removed, the cross-sectional shape of earlier that the rate at which the overlying ice the magma body is free to evolve under more can deform downwards is not likely to be more local forces; specifically, magma should begin to than a few mm s -1 . This explosive fragmentation concentrate into one or more structures resem- process is, of course, an excellent candidate for bling subaerial lava flows (Fig. 8a, b). The change the origin of sudden j6kulhlaup production. It is will happen because the energy losses due to Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 21

The corresponding sill was initially c. 1 km wide (a) (the same horizontal length as the feeder dyke) dyke outcrop and increased in thickness as it grew, the mean thickness reaching several metres after about 10 hours. Thus the thickness is largely irrelevant and the sill perimeter is somewhere between 1 and 2 km. The uncertainty arises because the base is always in contact with a stationary rock surface but the top has a layer of water between it and the stationary ice, thus making the frictional slip conditions more complicated. The same issue would apply to a subglacial lava flow, because ice sheet sill even if its top were in contact with the overlying ice, there would be a layer of low-viscosity water, however thin, at the interface. Again this hardly (b) matters, however, because even the conservative sill perimeter of c. 1 km is vastly greater than the worst case (2 x 18=)36m friction-generating perimeter of the flow. Changing the assumed slope down which the flow-like structure moves would change its cross-sectional shape somewhat (note the presence of sin o~ in equation (12)), but again not enough to change the fact that any small instability which causes the advance of the magma to become concentrated into one or more flow-like structures will be favoured. Water generated by heat transfer into the ice will tend to be channelled along the side(s) of the flow(s), (c) and the system will only remain stable as long as the pressure in the water is maintained high enough to suppress magma vesiculation to the point of fragmentation. If this occurs, one or more discrete lava flows will emerge from beneath the ice (Fig. 8b). However if instead fragmentation occurs, then the factors already discussed relating to subglacial explosive activity come into play (Fig. 7e, f), and new lava flow lobes will grow away from the dyke (Fig. 8c).

Fig. 8. The development of subglacial lava flow structures. (a) Sill reaches edge of ice sheet and elastic constraints are relaxed but no explosive fragmentation Summary of sill magma occurs; (b) water escapes and flow regin~e evolves to resemble that of subaerial flows. (1) With appropriate modifications, the princi- (e) Alternative source of subglacial lava flows formed ples used to analyze subaerial eruptions and when sill is explosively fragmented, dyke exit intrusions (both dyke- and sill-like) in silicate experiences atmospheric pressure, and flows are rocks can be applied to eruptions under, into and generated from a lava fountain over the vent through ice sheets, as illustrated in Figures 9 (see Fig. 7e, f). and 10. The geometries of dyke and sill emplacement and subsequent behaviour (decompression, transition to phreatomagmatic behaviour, etc.) friction decrease as the cross-sectional shape of a are very efficient at delivering heat to the sur- moving fluid body becomes more equant. Con- rounding ice and creating high volumes of melt- sider the comparison made earlier between the water early in the eruptions, perhaps accounting advance of a sill and a surface flow with the same for the production of major initial pulses of volume flux. The flow was 2m thick and 18 m meltwater sometimes observed in Icelandic erup- wide, thus having a total perimeter at right angles tions (e.g. Bj6rnsson 1992). to the direction of travel of 40m of which the (2) Typical basaltic magma densities and 18 m at the base is in contact with the ground. volatile contents are such that dykes which Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

22 L. WILSON & J. W. HEAD

SUBGLACIAL AND ENGLACIAL INTRUSIONS SUBAERIAL ENVIRONMENT ~ '~'- PLUME ICE LAYER/GLACIER PROPAGATES __~ TO SURFACE (~) SILL REACHES (~) DYKE MELTWATERLENS OR ICE CAVERN / s,,,

/ BEDROCK'

Fig. 9. Diagrammatic representation of subglacial and englacial intrusions. At (1) the dyke may become a sill at the bedrock-ice interface, and subsequent heating of the ice can lead to meltwater production or, if drainage occurs, an ice cavern and transition to a flow. In (2) the dyke propagates a significant distance into the overlying ice, which appears rheologically similar to the underlying silicates at these strain rates; if enough volatile exsolution occurs, propagation to the surface may occur and an eruption plume could be produced. Heating and ice melting at the dyke margin causes it to lose coherence and collapse to form a rubble pile. Such rubble piles could lie at the cores of hyaloclastite ridges.

KEY PHASES IN SUBGLACIAL ERUPTIONS ® ® ® (•)HAWAIIAN t OR SUBSIDENCE SURTSEYAN

ICE M~LTWATER SILL ' S

I[~BF'ROC~ ', /

Fig. 10. Diagrammatic representation of key phases of subglacial eruptions. At (1) dyke intrusion leads to sill formation at the bedrock-dyke interface; at (2) heating produces a meltwater lens. If meltwater is drained and ambient atmospheric pressure is reached, phreatomagmatic eruptions will occur, accompanied by subsidence and ice cauldron formation; At (3) collapse of the ice surface can lead to Hawaiian or Surtseyan eruptions, depending on the involvement of meltwater in the vent, at (4). Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 23

Table 4. Variation with time, t of the heat flux, q and the total amount of heat released so far, H, from a sill intruded under ice. Also given are the thicknesses of the cooled crust on the sill, de, the ice layer melted, di, and the layer of water produced, d,.

t q H dc di dw (s) (kW m -2) (MJ m -2) (m) (m) (m)

1 2395 4.8 0.0021 0.0156 0.0143 3 1383 8.3 0.0036 0.0270 0.0248 10 757 15.1 0.0066 0.0493 0.0452 30 437 26.2 0.0115 0.0854 0.0783 100 240 47.9 0.0210 0.1559 0.1430 300 138 83.0 0.0363 0.2710 0.2477 1000 76 151.5 0.0663 0.4931 0.4522 3 000 44 262.4 0.1149 0.8541 0.7832 10000 24 479.0 0.2098 1.5593 1.4299 30000 14 830.0 0.3630 2.7100 2.4770

would have reached the surface and erupted a long time scale and subsidence of overlying ice subaerially can, if they reach the surface under occurs to form an ice cauldron (Fig. 6). an ice sheet, penetrate 20 to 30% of the way (6) If magma supply continues after the sill through the ice and stall as dyke-like intrusions magma reaches the edge of the ice sheet, the (Figs 1-3). In most cases there would be no release of confining pressure can have several surface manifestation of these events other than consequences (Figs 7a-c & 8a-c). Rapid water possible minor subsidence, but in some cases gas release (j6kulhlaup formation) can occur, exa- venting, surface disturbance, and even minor cerbated by the explosive decompression of sill phreatomagmatic activity might be observable. magma and enhanced heating of the overlying Subsequent ice melting will render these intru- ice (Fig. 7d). A subglacial lava fountain will form sions unstable and they will collapse to form over the feeder dyke, locally greatly increasing characteristic fragmental deposits at the base of the ice melting rate (Fig. 7e), and a new sub- the ice. glacial lava flow or group of flows (Figs 7e, f & (3) Sills can form at the bases of ice sheets 8c) will form, the ice-melting efficiency of which (Fig. 4). The pressures in the magmas in these will be enhanced if overlying ice collapses into sills will typically be c. 0.5 MPa higher than the the cavity vacated by disruption of the initial sill lithostatic pressure of the overlying ice (Table 1) (Fig. 7f). Alternatively, if explosive decompres- and at low magma water contents exsolution of sion of the sill does not occur, the shape of the mainly CO2 will cause the sills to have vesicula- subglacial sill may evolve into that of one or rities typically ranging from 10% (up to 2 km ice more lava flow-like structures (Fig. 8c). cover) to 30% (a few hundred metres ice cover). (7) A wide array of volcanic landforms has Under shallower ice depths and with high been observed on Mars (Hodges & Moore 1994). magma water contents (Table 2), enough water Application of the principles developed here to exsolution may occur that spontaneous magma Mars provides criteria to assess possible examples fragmentation takes place and sills may be of intrusion and eruption below polar deposits, intruded largely as hyaloclastite deposits. Such ice fields, and glaciers (Garvin et al. 2000; Ghatan intrusions can reach lateral extents of c. 1 km and & Head 2001; Head & Wilson 2002). thicknesses of 1-2 metres in c. 1 hour (Table 3). Discussions in the field with Magnus Gudmunds- (4) Comparison of the typical rates of increase son, Snorri Snorrason, Elsa Vilmundardottir, Sveinn of thickness of subglacial sills (Table 3) with the Jakobsson, J. Smellie and I. Skilling are gratefully rate of growth of chilled crust as they interact acknowledged. Comments on the manuscript by J. with overlying ice (Table 4, Fig. 5) shows that Smellie and two anonymous reviewers helped us to cooling will almost never inhibit their emplace- clarify a number of issues. We thank A. C6t6 for help ment; intrusion will continue until either the in drafting. This paper is based on an invited presen- tation given at the Volcano/Ice Interaction meeting in magma supply ceases or the sill reaches the edge Reykjavik, Iceland, in August, 2000. We gratefully of the ice sheet. acknowledge financial support from NASA through (5) If magma supply ceases before the sill the Planetary Geology and Geophysics Program and magma reaches the edge of the ice sheet, all of the the Mars Data Analysis Program, and from PPARC available heat is extracted from the magma over through grant PPA/G/S/2000/00521. Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

24 L. WILSON & J. W. HEAD

Appendix di thickness of ice melted adjacent to sill (m) ds thickness of sill near feeder dyke (m) A vertical extent of dyke (m) dw thickness of water layer produced by ice D thickness of subaerial lava flow (m) melting (m) E horizontal extent of sill on either side of e constant used in equation (5b), equal to feeder dyke (m) [pm/(2Kc)] (kg 2 m -4 s -2) H amount of heat released by magma per unit f constant used in equation (5b), equal to area of ice contact (J m -2) [(pmb)/(2Kch)] (kg 2 m -4 s -z) Jc constant in CO2 solubility law, equal to g acceleration due to gravity, equal to 9.8 3.4 × 10 -6 (dimensionless) (ms -2) Kc constant in CO2 solubility law, equal to h constant used in equation (5b), equal to 6 × 10 -12 (Pa -1) [(b 2 - 4ac) 1/2] (kg mol -l) Kw constant in water solubility law, equal to k m thermal conductivity of solidified magma, 6.8 x 10 -8 (Pa -°7) equal to 3.1 (W m-l K-l) L horizontal extent of dyke (m) mc molecular weight of CO2, equal to 43.99 L~ latent heat of fusion of ice, equal to (kg kmol-l ) 3.3 × 105 (Jkg -1) n weight fraction of water dissolved in basalt Lm latent heat of fusion of magma, equal to (dimensionless) 2.09 × 105 (Jkg -l) nc solubility of CO2 in basalt (dimensionless) P ambient pressure (Pa) ne weight fraction of CO2 exsolved from Pa atmospheric pressure, equal to c. 105 (Pa) magma (dimensionless) Pc pressure in static magma column extending nt total CO2 content of magma (dimension- from reservoir to ice-rock interface (Pa) less) Pe pressure in dyke tip in excess of local nw solubility of water in basalt (dimensionless) lithostatic load (Pa) q heat loss rate per unit area of magma-ice Pi magma pressure at ice-rock interface (Pa) contact (W m -2) Ppt pressure in dyke tip while dyke is propa- r radius of gas bubble (m) gating (Pa) t time (s) Pr pressure in magma at roof of magma u rise speed of gas bubbles through magma reservoir (Pa) (ms -l ) Ps magma pressure at sill inlet from feeder x depth of upper dyke tip below ice surface dyke (Pa) (m) Pt residual pressure in dyke tip after it comes y thickness of surface ice layer (m) to rest (Pa) z depth of magma reservoir below rock-ice Q universal gas constant, equal to 8.314 interface (m) (kJ kmol- 1 K- l) oL slope of ground under subaerial lava flow Tm magma temperature, equal to 1473 (K) (degrees) Tw temperature of meltwater above chilled sill /3 bulk density of magma (kg m -3) crust, equal to 277 (K) tim mean bulk density of magma in dyke UL flOWspeed of subaerial lava (m s -1) (kgm -3) UM rise speed of magma in dyke (ms -l) ~]m viscosity of magma in dyke, equal to 100 Us speed of magma flowing into sill (m s -1) (Pa s) V volume flux of magma flowing though dyke 7/t~ viscosity of magma in subaerial flow, equal (m3s -1) to 50 (Pa s) W mean width of dyke (m) ni thermal diffusivity of ice, equal to c. 10 -6 Y yield strength of subaerial lava, equal to (m2s -1) 700 (Pa) nm thermal diffusivity of solidified magma, a constant used in equation (5b), equal to equal to c. 10 -6 (m 2 s -1) [pmQTm(nt - Jc)] (kg 2 m -l s -2 mo1-1) A constant in heat transfer equation, equal to b constant used in equation (5b), equal to 1.1514 (dimensionless) [mc(1 - nt + Jc) - pmQTmKc] (kgmo1-1) # shear modulus of crustal rocks, equal to c constant used in equation (5b), equal to 3 x 109 (Pa) [mcKc] (m s 2 mol-1 ) u Poisson's ratio of crustal rocks, equal to Cm specific heat of solidified magma, equal to 0.25 (dimensionless) 1200 (Jkg -1K -~) p density of subaerial lava flow, equal to d distance penetrated by thermal changes due 1000 (kg m -3) to conduction (m) pc density of CO2 gas (kg m -3) dc thickness of chilled crust on sill (m) pi density of ice, equal to 917 (kgm -3) Downloaded from http://sp.lyellcollection.org/ by guest on September 26, 2021

THERMAL CONSEQUENCES OF SUBGLACIAL ERUPTIONS 25

Pm density of basaltic magmatic liquid, equal Hawaii. Journal of Geophysical Research, 91, to 2700 (kgm -3) 2177-2185. Pr density of crustal rocks, equal to 2300 GHATAN, G. J. & HEAD, J. W. 2001. Candidate sub- (kgm -3) glacial volcanoes in the south polar region of Mars. Lunar and Planetary Science, 32, #1039 Og gas density in bubble (kgm -3) (CD ROM). GLEN, J. W. 1952. Experiments on the deformation of References ice. Journal of Glaciology, 2, 111-114. GRGNVOLD, K. & JOHANNESSON, H. 1983. Eruption in ALLEN, C. C. 1979. Volcano-ice interactions on Mars. Grimsv6tn 1983, course of events and chemical Journal of Geophysical Research, 84, 8048-8059. studies of the tephra. J6kull, 34, 1-11. ALLEN, C. C. 1980. Icelandic subglacial volcanism: ther- GUDMUNDSSON, m. 1987. Lateral magma flow, caldera mal and physical studies. Geology, 88, 108-117. collapse, and a mechanism of large eruptions in BJt3RNSSON, H. 1975. Subglacial water reservoirs, j6k- Iceland. 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